Recall from Remark REF  that the knapsack variant, where we ask for a solution in \(\mathbb {Z}^k\)  (let us call this variant
integer-valued knapsack) can be reduced to our version of knapsack,
where the solution is restricted to \(\mathbb {N}^k\) . Hence, there might be groups where integer-valued knapsack is easier than
standard knapsack. Indeed, in the introduction we remarked
that integer-valued knapsack for the group \(\mathbb {Z}\)  with binary encoded integers
can be solved in polynomial time. This motivates the investigation of integer-valued knapsack for other
graph groups. In [1]} we proved that integer-valued knapsack for \(\mathbb {G}(\mathsf {C4})\)  is NP-complete,
even if group elements are given by uncompressed words. On the other hand, our NP-hardness proof for \(\mathbb {G}(\mathsf {P4})\) 
makes essential use of the fact that solution vectors are restricted to \(\mathbb {N}^k\) , and it remains open whether integer-valued knapsack for
\(\mathbb {G}(\mathsf {P4})\)  is NP-hard. In this context, it is interesting to note that there are some interesting differences between
\(\mathbb {G}(\mathsf {P4})\)  and \(\mathbb {G}(\mathsf {C4})\)  with respect to algorithmic problems. For instance, whereas the subgroup membership
problem for \(\mathbb {G}(\mathsf {C4})\)  is undecidable [2]}, the subgroup membership problem for \(\mathbb {G}(\mathsf {P4})\)  (and any graph group
\(\mathbb {G}(\Gamma )\)  with \(\Gamma \)  chordal) is decidable [3]}, [4]}.
